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Monty Hall Problem
I was reading the article by Michael Shermer
on page 40 of the October 2008 issue of Scientific American.
The following text is copied directly from that article.
Imagine that you are a
contestant on the classic television game show Let's Make a
Deal.
Behind one of 3 doors is a brand-new automobile. Behind the
other two
are goats. You choose door number one. Host Monty
Hall, who knows
what is behind all three doors, shows you that a goat is behind number
two, then inquires: Would you like to keep the door you chose
or
switch? Our folk numeracy--our natural tendency to think
anecdotally
and to focus on small number runs--tells us that it is 50-50, so it
doesn't matter, right?
Wrong. You had a one in three chance to
start, but now that Monty has shown you one of the losing doors, you
have a two-thirds chance of winning by switching.....
The
article goes on to discuss this problem further, but I failed to find
the explanation convincing. The article references the
discussion of
this problem in the book The Drunkards's Walk by
Leonard Mladinow and it credits Marilyn vos Savant for first presenting
this puzzle in her Parade magazine column in 1990.
The
Michael Shermer article continues with other subjects, but I didn't
read further. I stopped and thought. I had read
often of this problem
previously. In fact, I had read Marilyn vos Savant's
discussions of it
a number of times. But, every time, I had failed to
understand this
"simple" problem.
Recently, I read "Outliers" by Malcolm
Gladwell. In it he talks about the importance of persistence
in
understanding mathematics. I've always thought of myself at
being very
good at math, including the basic concepts of probability.
Yet, I had
not understood the answer to this problem. Rather than
continuing to
read the rest of Michael Shermer's article and Scientific American, I
took the advice that I had gotten from Malcolm Gladwell's book, and I
determined to persevere until I inderstood the answer to this problem.
I
analyzed the problem multiple ways, as I had done in the past, but I
kept coming up with the wrong answer (that there was an equal 50%
chance of winning if I stayed with the present door or if I
switched).
This is the same wrong answer given to Marilyn vos Savant by numerous
experts, including a number of mathematicians.
About 15
minutes later, I finally convinced myself that I understood how to
obtain the correct answer to this problem. I wrote up these
thoughts a
few days later while they were still fresh in my mind.
I
recognize that my solution is very basic and very obvious. It
has, of
course, been thought of by Marilyn vos Savant, Michael Shermer, Leonard
Mlodinow and multitudes of others. But still, I wanted to
share it and
to share how I got to it. In hindsight, I realize that
Malcolm
Gladwell's book also motivated me to share these thoughts.
I
intentionally avoided researching the problem on the web. I
wanted to
solve it myself. I've since done a Google search and found
much information on this and related problems. For your
convenience, I've also included some Google links on this page..
Herman Held
My Solution
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